A fatwa is issued by a recognized religious authority in Islam. But since there is no hierarchical priesthood or anything of the sort in Islam, a fatwa is not necessarily "binding" on the faithful. The people who pronounce these rulings are supposed to be knowledgable, and base their rulings in knowledge and wisdom. They need to supply the evidence from Islamic sources for their opinions, and it is not uncommon for scholars to come to different conclusions regarding the same issue.
As Muslims, we look at the opinion, the reputation of the person giving it, the evidence given to support it, and then decide whether to follow it or not. When there are conflicting opinions issued by different scholars, we compare the evidence and then choose the opinion to which our God-given conscience guides us.